# Bismut's Formula and Its Generalizations

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# Default probabilities, credit derivatives, and computational issues.

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The two main approaches to modeling defaults, structural and intensity based, will be reviewed. We show that perturbation methods are useful in approximating default probabilities in the context of stochastic volatility models. We then consider the case of many names and we discuss various ways of creating correlation of defaults. In highly-dimensional models, Monte Carlo simulations remain a powerful tool for computing prices of credit derivatives such as CDO's tranches and associated greeks. We propose an interactive particle system approach for computing the small probabilities involved in these financial instruments.

# Diffusions in random environment and ballistic behavior.We study diffusions in random environment in higher dimensions. We assume that the environment is stationary and obeys finite range dependence.

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We study diffusions in random environment in higher dimensions. We assume that the environment is stationary and obeys finite range dependence. Once the environment is chosen, it remains fixed in time. To restore some stationarity, it is common to average over the environment. One then obtains the so-called annealed measures, that are typically non-Markovian measures.

Our goal is to study the asymptotic behavior of the diffusion in random environment under the annealed measure, with particular emphasis on the ballistic regime

('ballistic' means that a law of large numbers with non-vanishing limiting velocity holds). In the spirit of Sznitman, who treated the discrete setting, we introduce

conditions (T) and (T'), and show that they imply, when d>1, a ballistic law of large numbers and a central limit theorem with non-degenerate covariance matrix.

As an application of our results, we highlight condition (T) as a source of new examples of ballistic diffusions in random environment.

# Large deviations for partition functions of directed polymers and other models in an IID field.

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Consider the partition function of a directed polymer in an IID

field. Under some mild assumptions on the field, it is a well-known fact

that the free energy of the polymer is equal to some deterministic constant

for almost every realization of the

field and that the upper tail large deviations is exponential. In this

talk I'll discuss the lower tail large deviations and present a method

for estimating it. As a consequence, I'll show that the lower

tail large deviations exhibits three regimes, determined by the

tail of the negative part of the field. The method applies to other

oriented models and can be adapted to non-oriented models as well. This

work extends the results of a recent paper by Cranston Gautier and

Mountford. A preprint is availabe on www.math.uci.edu/~ibenari

# Asymptotic Enumeration of Spanning Trees via Traces and Random Walks

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Methods of enumeration of spanning trees in a finite graph and relations to

various areas of mathematics and physics have been investigated for more

than 150 years. We will review the history and applications. Then we will

give new formulas for the asymptotics of the number of spanning trees of a

graph. A special case answers a question of McKay (1983) for regular

graphs. The general answer involves a quantity for infinite graphs that we

call ``tree entropy", which we show is a logarithm of a normalized

[Fuglede-Kadison] determinant of the graph Laplacian for infinite graphs.

Proofs involve new traces and the theory of random walks.

# Lyapounov norms for random walks in a random potential.

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Abstract : We consider a simple random walk on Z

, d > 3. We also consider

a collection of i.i.d. positive and bounded random variables ( V? (x) )x?Z d , which will

serve as a random potential. We study the annealed and quenched cost to perform

long crossings in the random potential ? + ? V? (x), where ? is positive constant

and ? > 0 small enough . These costs are measured by the Lyapounov norms We

prove the equality of the annealed and the quenched norm. We will also discuss the

relation between the Lyapounov norms and the path behavior of the random walk

in the random potential.

# Don't love your children equally: the advantages of asymmetric damage segregation in fissioning organisms.

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A number of recent experiments have shown that several organisms

that reproduce by fissioning (e.g. E. coli bacteria)

don't share the cellular damage they have

accumulated during their lifetime equally among their offspring. Using

a stochastic PDE model, David Steinsaltz and I have shown that under quite

general conditions the optimal asymptotic growth rate for a population

of fissioning organisms is obtained when there is a non-zero but moderate

amount of preferential segregation of damage -- too much or too little

asymmetry is counter-productive. The proof uses some new results of ours

on quasi-stationary distributions of one-dimensional diffusions and

some Sturm-Liouville theory. The talk is intended for a probability

audience and I won't assume any knowledge of biology.

# Don't love your children equally: the advantages of asymmetric damage segregation in fissioning organisms.

## Speaker:

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A number of recent experiments have shown that several organisms

that reproduce by fissioning (e.g. E. coli bacteria)

don't share the cellular damage they have

accumulated during their lifetime equally among their offspring. Using

a stochastic PDE model, David Steinsaltz and I have shown that under quite

general conditions the optimal asymptotic growth rate for a population

of fissioning organisms is obtained when there is a non-zero but moderate

amount of preferential segregation of damage -- too much or too little

asymmetry is counter-productive. The proof uses some new results of ours

on quasi-stationary distributions of one-dimensional diffusions and

some Sturm-Liouville theory. The talk is intended for a probability

audience and I won't assume any knowledge of biology.

# Horseshoes in Multidimensional Scaling

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Classical multidimensional scaling (MDS) is a method for visualizing high-dimensional point clouds by mapping to low-dimensional Euclidean space. This mapping is defined in terms of eigenfunctions of a matrix of interpoint proximities. I'll discuss MDS applied to a specific dataset: the 2005 United States House of Representatives roll call votes. In this case, MDS outputs 'horseshoes' that are characteristic of dimensionality reduction techniques. I'll show that in general, a latent ordering of the data gives rise to these patterns when one only has local information. That is, when only the interpoint distances for nearby points are known accurately. Our results provide insight into manifold learning in the special case where the manifold is a curve. This work is joint with Persi Diaconis and Susan Holmes.